Partial differential equations (Winter Semester 2004/05)
- Lecturer: Professor Guido Schneider
- Classes: Lecture (01058), Problem class (01059)
- Weekly hours: 4+2
- Audience: Mathematicians, physicians, electrical engineering,... (4.-9. semester)
Almost all rules of physics and engineering, many rules in life sciences and economics are formulated as ordinary or partial differential equations. As different as these applications of differential equations are, as different the behavior of their solutions. Therefore, a mathematical theory which wants to cover all differential equations can only cover the absolute basics. Important examples will play the same role as a good mathematical theory for PDEs. We will discuss:
* fundamental examples: Laplace equation, heat equation, first order systems, shocks
* functional analytic tools: Fourier transform, Sobolev spaces, inequalities
* linear and nonlinear PDEs on the real line: transport, diffusion, dispersion, Duhamel's formula, local existence and uniqueness, examples,
* linear and nonlinear PDEs with boundary conditions: periodic, Dirichlet, Neumann, mixed boundary conditions, sectorial operators, spaces of fractional powers, local existence and uniqueness, examples
* the Millenium-problem: The global existence of smooth solutions of the 3D Navier-Stokes problem.
| Schedule | |||
|---|---|---|---|
| Lecture: | Monday 9:45-11:15 | Seminarraum 31 | Begin: 18.10.2004, End: 16.2.2004 |
| Wednesday 11:30-13:00 | Seminarraum 34 | ||
| Problem class: | Thursday 14:00-15:30 | Seminarraum 32 | Begin: 21.10.2004, End: 17.2.2005 |
| Lecturers | ||
|---|---|---|
| Lecturer, Problem classes | Professor Guido Schneider | |
| Office hours: | ||
| Room Kollegiengebäude Mathematik (20.30) | ||
| Email: | ||
Contents
- fundamental examples: Laplace equation, heat equation, first order systems, shocks
- functional analytic tools: Fourier transform, Sobolev spaces, inequalities
- linear and nonlinear PDEs on the real line: transport, diffusion, dispersion, Duhamel's formula, local existence and uniqueness, examples,
- linear and nonlinear PDEs with boundary conditions: periodic, Dirichlet, Neumann, mixed boundary conditions, sectorial operators, spaces of fractional powers, local existence and uniqueness, examples
- the Millenium-problem: The global existence of smooth solutions of the 3D Navier-Stokes problem.
References
* John, Fritz: Partial Differential Equations; Springer, New York, Heidelberg, 4. Aufl., 1991
* Renardy, Michael: Rogers, Robert C. An introduction to partial differential equations. Second edition. Texts in Applied Mathematics, 13. Springer-Verlag, New York, 2004. xiv+434 pp. ISBN: 0-387-00444-0
* Evans, Lawrence C.: Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. xviii+662 pp. ISBN: 0-8218-0772-2
* Henry, Daniel: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981
* Temam, Roger: Infinite-dimensional dynamical systems in mechanics and physics. Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
* Strauss, Walter A.: Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. x+91 pp. ISBN: 0-8218-0725-0